Although a SOT-23-6 MCU is the obvious approach, if you want to avoid the toolchain and coding work involved in using one, then the following timing diagram illustrates one approach:
pwr is a power-on flip-flop that is clocked by clk. It is reset by a power-on event.
long is a one-shot with a long time period that represents your long-...
You're operating the device outside its RECOMMENDED OPERATING CONDITIONS. Of course you're free to do that but this can lead to unspecified behavior.
In your case the "Input Rise and Fall Time" specification is being exceeded. For 5V operation the specification is 5 ns/V MAX. So you need to ensure that your CP input rises from 0 to (0.7 * Vcc) ...
I suspect that the slow rising clock edge might cause timing problems. I suggest that you swap R106 and C109. That will give you a fast rising clock edge on the clock when the button is pressed. I think a slow falling edge is less likely to cause problems, because Q and Qbar won't be changing at the same time.
I think the best way to modify your schematic to do what you want is to throw away most of the parts and use LTC2954 push button controller. It is designed exactly for one button control, does debouncing for you, supports adjustable power on and power down times and works up to 26V. Oh, and it does not assert output when you first apply power to it.
When Latch Enable (LE) is high, the data in the shift register will immediately appear on the LEDs. So, you don't want it high while you are loading the shift register or the LEDs might flicker.
#1 is better than #2, but I would do it this way.
Start with LE low. Load the shift register, then set LE high momentarily. Keep it low until after you have loaded ...
You need a better method of debouncing the switch than the RC.
The RC may be good enough if you insert a Schmitt trigger gate after the RC and before the clock input eg. SN74LVC1G17-Q1.
CMOS gates have poorly characterized analog parameters- the minimum hysteresis is 560mV and the maximum 1320mV, but if the bounce time of the switch is less than a few ms and ...